Complex numbers – Number system extending real numbers by adding an imaginary part.Orbit – Sequence of numbers arising from iterating an escape-time fractal process on a given input (the path this number takes).Escape-time fractal – Fractals that arise by iterating a function on some input, usually a complex number, and considering it escaped if its magnitude exceeds some threshold after a fixed iteration count.Fractal – Some kind of object, which exhibits self-similarity at different scales.If you really want to get a retro feel, then you can go to Sprott's Fractal Gallery, where a new fractal is automatically generated every day! The best place for everything fractal is Fractalforums ( historical site). Popular software for fractal rendering includes Mandelbulber, Mandelbulb 3D, Fragmentarium. These will be the fractals whose construction and rendering we will focus on in this topic. The famous Mandelbrot fractal is an escape-time fractal. Escape-time fractals – These are systems that recursively iterate over the input and try to find if the result stays within some bound (does not escape in some certain number of iterations).Strange attractors – Systems of differential equations that evolve in time can create fractal-like shapes.Ie, branches of a tree have a similar structure as the tree itself. The whole process of iterating the systems inherently creates a fractal structure as the same rules are applied at each iteration to grow the tree. You can think of a simple tree generated by an L-system. Lindenmayer-systems – As you might have guessed, the L-systems we learned in the Computer Graphics course are also fractals.Other commonly known examples are the Sierpiński triangle and the Menger sponge. ![]() For example, the Koch snowflake is constructed by drawing new triangles in the middle of the outer edges of existing triangles, thus extending the shape.
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